Electromagnetic Waves

Step 1: Understanding the group velocity.
The group velocity is given by . Using the given dispersion relation , we find . Differentiating with respect to , we get . The result shows that the group velocity is inversely proportional to the phase velocity . Hence, the correct answer is option (B).

Step 1: Understanding the skin depth.
The skin depth is related to the wave number in the metal by the formula: where is the complex wave number, and is the imaginary part representing the attenuation of the wave in the metal.
Step 2: Finding the imaginary part.
Given that the wave number is , the imaginary part is 750. The skin depth is: Step 3: Conclusion.
Thus, the skin depth of the wave in the metal is 1.33 mm.

Step 1: Understanding the wave equation.
The given electric field corresponds to an electromagnetic wave with the general form . Comparing the arguments of the sine functions, we see that and . By using the relation , we can solve for . After simplifying, we get .
Step 2: Conclusion.
Thus, the correct answer is option (A), .

Step 1: Understanding the situation.
In the given setup, a plane electromagnetic wave is incident normally on an interface at separating two media with different wave velocities and . For the reflected wave, the electric field vector will reverse direction, and its wave vector will change direction as well. The general form of the electric field in the second medium (after reflection) will be the same as the incident wave but with a reversed sign in the exponential factor.
Step 2: Identifying the correct reflected field.
The electric field in the reflected wave should have the same magnitude but an opposite direction. Thus, the reflected electric field vector will be:
Step 3: Conclusion.
Thus, the correct answer is (C), where the reflected wave has the electric field vector in the opposite direction with the same magnitude but the opposite sign in the exponential term.

Step 1: Determine the magnetic field.
For an electromagnetic wave propagating along -axis, Step 2: Calculate the Lorentz force.
At : Since : Step 3: Total force.
Step 4: Final Answer.
The instantaneous force is

Step 1: Use Maxwell’s equation.
From Faraday’s law, Step 2: Compute curl of .
Step 3: Relate to .
Integrate w.r.t. time: Step 4: Final Answer.
Hence,